- #1

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- Homework Statement
- The Zeta function is defined as $$\zeta(n) \ = \ \sum _{k=1} ^{\infty} \frac{1}{k^n}.$$ Given that ##\zeta(4) = \pi^4/90##, estimate a value of ##\pi## correct to three significant figures. How many terms are needed in the series for this?

- Relevant Equations
- Given in the question.

I have solved questions where I have been asked to find something correct to ##m## places of decimal using some series. See this thread. The logic was, the program would terminate at the ##k##th term if the ##k+1##th term is ##<10^{-m}##.

But how do I terminate the series here? Say I calculate ##S_{k+1}## and ##S_k##, and find a difference ##t_{k+1}## (actually the ##k+1##th term). Notwithstanding anything else, this term can be approximated to any number of significant terms. For example, if ##t_{k+1} \ = \ 0.0000000001,## I can still approximate that as ##0.0000000001000## and conclude that it has four significant figures. In fact, the trailing zeros can be anything depending on how precisely my computer can calculate. What should my condition of termination be in this case? More specifically, how do I check whether the current sum is

But how do I terminate the series here? Say I calculate ##S_{k+1}## and ##S_k##, and find a difference ##t_{k+1}## (actually the ##k+1##th term). Notwithstanding anything else, this term can be approximated to any number of significant terms. For example, if ##t_{k+1} \ = \ 0.0000000001,## I can still approximate that as ##0.0000000001000## and conclude that it has four significant figures. In fact, the trailing zeros can be anything depending on how precisely my computer can calculate. What should my condition of termination be in this case? More specifically, how do I check whether the current sum is

*correct*up to certain significant figures?